Math 2.0

Differential geometry Definition

Differentiable curve

Builds on Parametrised curveSmoothness classesInterval

Definition

Let n≥1n \ge 1 be an integer, let k≥1k \ge 1 be an integer or k=∞k = \infty, and let I⊆ℝI \subseteq \R be an interval with more than one point. Let γ ⁣:I→ℝn\gamma \colon I \to \R^n, γ(t)=(γ1(t),…,γn(t))\gamma(t) = (\gamma_1(t), \dots, \gamma_n(t)), be a parametrised curve. The curve γ\gamma is a differentiable curve of class CkC^k if each coordinate function γi\gamma_i is of class CkC^k (see smoothness classes) on II. This means that γi\gamma_i is kk times differentiable and its derivatives up to order kk are continuous. At an end point of II that belongs to II, the derivatives are one-sided. A smooth curve is a differentiable curve of class C∞C^\infty.

Remarks

Example: γ(t)=(t,t2)\gamma(t) = (t, t^2), t∈ℝt \in \R, is a smooth curve, since both coordinates are polynomials. Non-example: γ(t)=(t,∣t∣)\gamma(t) = (t, \abs{t}), t∈ℝt \in \R, is a continuous curve but not of class C1C^1, because the second coordinate has no derivative at t=0t = 0. A differentiable curve can still have a cusp in its image: t↦(t2,t3)t \mapsto (t^2, t^3) is smooth, but its velocity vanishes at t=0t = 0. Curves whose velocity never vanishes are the regular curves.

Reading path

To understand this page from the ground up, read these 25 pages in order. The foundations come first.

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Construction of the reals from Cauchy sequences, concept, no evidence levelConstruction of the r…conceptEuclidean space, object, no evidence levelEuclidean spaceobjectFunction, concept, no evidence levelFunctionconceptIntegers, object, no evidence levelIntegersobjectOrdered field, definition, no evidence levelOrdered fielddefinitionRational numbers, object, no evidence levelRational numbersobjectSmoothness classes, concept, no evidence levelSmoothness classesconceptFréchet derivative, definition, no evidence levelFréchet derivativedefinitionFunction of several variables, concept, no evidence levelFunction of several v…conceptVector-valued function, concept, no evidence levelVector-valued functionconceptRing of integers, statement, no evidence levelRing of integersstatementLimit of a function of several variables, concept, no evidence levelLimit of a function o…conceptThe rationals form a field, statement, no evidence levelThe rationals form a …statementContinuity in R^n, definition, no evidence levelContinuity in R^ndefinitionDifferentiable function of several variables, definition, no evidence levelDifferentiable functi…definitionPartial derivative, definition, no evidence levelPartial derivativedefinitionThe rationals form an ordered field, statement, no evidence levelThe rationals form an…statementDifferentiable on an open set, definition, no evidence levelDifferentiable on an …definitionOrder on the reals, definition, no evidence levelOrder on the realsdefinitionHigher partial derivatives, definition, no evidence levelHigher partial deriva…definitionIterated derivative, definition, no evidence levelIterated derivativedefinitionInterval, definition, no evidence levelIntervaldefinitionC^k function characterised by continuous derivatives, statement, no evidence levelC^k function characte…statementParametrised curve, definition, no evidence levelParametrised curvedefinitionDifferentiable curve, definition, no evidence levelDifferentiable curvedefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Construction of the reals from Cauchy sequences concept
  2. Euclidean space object
  3. Function concept
  4. Integers object
  5. Ordered field definition
  6. Rational numbers object
  7. Smoothness classes concept
  8. Fréchet derivative definition
  9. Function of several variables concept
  10. Vector-valued function concept
  11. Ring of integers statement
  12. Limit of a function of several variables concept
  13. The rationals form a field statement
  14. Continuity in R^n definition
  15. Differentiable function of several variables definition
  16. Partial derivative definition
  17. The rationals form an ordered field statement
  18. Differentiable on an open set definition
  19. Order on the reals definition
  20. Higher partial derivatives definition
  21. Iterated derivative definition
  22. Interval definition
  23. C^k function characterised by continuous derivatives statement
  24. Parametrised curve definition
  25. Differentiable curve definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Differentiable_curve
About this page
TagX66ZPDYAKC
URIm2://X66ZPDYAKC
Kinddefinition
WritingW0 stub
MSC53A04
DefinesDifferentiable curve
UsesParametrised curve, Smoothness classes, Interval
Changed2 November 2026

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